Sample Exam Solutions

pdf

School

University of Houston *

*We aren’t endorsed by this school

Course

4364

Subject

Mechanical Engineering

Date

Oct 30, 2023

Type

pdf

Pages

Uploaded by ibrahimmriaz

1 MECE 4364 Heat Transfer Midterm Examination Solutions 1. ( 25 points ) A composite wall has three layers of materials, A, B and C. We know the thermal conductivities of two materials, 𝑘 ? = 20 𝑊/𝑚? and 𝑘 ? = 50 𝑊/𝑚? . The thicknesses are ? ? = 0.3 𝑚 , ? ? = ? ? = 0.15 𝑚 . The left surface of the wall is exposed to a hot air flow ( T ∞ = 800°C and h = 25 W/m 2 K). Under steady-state conditions, we find the left surface temperature of 𝑇 ?,𝑖 = 600℃ and the right surface temperature 𝑇 ?,𝑂 = 20℃ . a. ( 10 points ) Draw the thermal network of the system and mark all the thermal resistances (including the expressions). b. ( 5 points ) What is the heat flux through the composite wall? c. ( 10 points ) What is the thermal conductivity of material B, 𝑘 ? ?

3 2. ( 55 points ) As shown in the following figure, we have a plane wall of thickness L = 1 m and thermal conductivity 𝑘 = 2 𝑊/𝑚? . Due to internal heating, the wall experiences uniform heat generation at a rate of 𝑞̇ = 20 𝑊/𝑚 3 . The left surface of the wall is exposed to the ambient air ( T ∞ = 20°C and h = 10 W/m 2 K), and it also receives solar irradiation in the amount of G s = 200 W/m 2 . The right surface of the wall is perfectly insulated. Meanwhile, the surface is irradiating to the universe of 𝑇 ???? = 3 ? . The solar absorptivity and emissivity of the surface are α = ε = 0.9 8. a. ( 5 points ) What is the boundary condition at the right surface ( 𝑥 = ? )? Write down the mathematical expression. b. ( 10 points ) Assuming the temperature of the left surface is T s , write down the surface energy balance equation for the left boundary ( 𝑥 = 0). Note: Do not calculate anything, just the symbolic expression is fine. c. ( 15 points ) Calculate the steady state surface temperature T s for the given conditions. Note: 1) This is a problem unrelated to Part b, so do not try to get the result directly from your answer to Part b; 2) The Stefan-Boltzmann constant is 𝜎 = 5.67 × 10 −8 𝑊/𝑚 2 ? 4 ; and 3) You may need to do trial-and-error when calculating T s . d. ( 10 points ) Assuming steady state, what is the conduction equation that governs heat transfer in the wall? e. ( 10 points ) Now you have the conduction equation and the two boundary conditions, solve the equation and find the surface temperature 𝑇(𝑥 = ?) . f. ( 5 points ) What is the conduction heat flux at the surface 𝑥 = 0 at steady state?

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

4 Solution: a. The right surface ( 𝑥 = ? ) is perfected insulated 𝑑𝑇 𝑑𝑥 = 0 b. The energy balance equation applied at the left surface is 𝐸 ̇ 𝑖? = 𝐸 ̇ ??? 𝐸 ̇ 𝑖? = ℎ(𝑇 ∞ − 𝑇 ? ) + 𝛼𝐺 ? 𝐸 ̇ ??? = −𝑘 𝑑𝑇 𝑑𝑥 | 𝑥=0 + 𝜀𝜎(𝑇 ? 4 − 𝑇 ???? 4 ) Thus, the final form of the equation is ℎ(𝑇 ∞ − 𝑇 ? ) + 𝛼𝐺 ? = −𝑘 𝑑𝑇 𝑑𝑥 | 𝑥=0 + 𝜀𝜎(𝑇 ? 4 − 𝑇 ???? 4 ) c. Apply the energy conservation equation to the wall, we have 𝐸 ̇ 𝑖? − 𝐸 ̇ ??? + 𝐸 ̇ 𝑔 = 0 where 𝐸 ̇ 𝑖? = ℎ(𝑇 ∞ − 𝑇 ? ) + 𝛼𝐺 ? 𝐸 ̇ ??? 𝜀𝜎(𝑇 ? 4 − 𝑇 ???? 4 ) 𝐸 ̇ 𝑔 = 𝑞̇? ℎ(𝑇 ∞ − 𝑇 ? ) + 𝛼𝐺 ? − 𝜀𝜎(𝑇 ? 4 − 𝑇 ???? 4 ) + 𝑞̇? = 0 10 × (293 − 𝑇 ? ) + 0.98 × 200 − 0.98 × 5.67 × 10 −8 (𝑇 ? 4 − 3 4 ) + 20 × 1 = 0 we can solve for 𝑇 ? : 𝑇 ? = 487.7 ? = 214.7℃ d. At steady state, the conduction equation becomes 𝑘 𝑑 2 𝑇 𝑑𝑥 2 + 𝑞̇ = 0 e. Consider the two boundary conditions 𝑇| 𝑥=0 = 𝑇 ? 𝑑𝑇 𝑑𝑥 | 𝑥=𝐿 = 0 The solution is 𝑇(𝑥) = − 𝑞̇ 2𝑘 𝑥 2 + 𝑞̇? 𝑘 𝑥 + 𝑇 ?

5 The temperature at the right surface is then 𝑇(?) = 𝑞̇ 2𝑘 ? 2 + 𝑇 ? = 20 2 × 2 1 2 + 214.7 = 219.7℃ f. The conduction heat flux at the left surface 𝑞 ???? ′′ = −𝑘 𝑑𝑇 𝑑𝑥 | 𝑥=0 = −𝑞̇? = −20 𝑊/𝑚 2